computer simulations and crossover equation of state - inside mines

Fluid Phase Equilibria 200 (2002) 121–145

Computer simulations and crossover equation of stateof square-well fluids

S.B. Kiseleva,∗, J.F. Elya, L. Lueb, J.R. Elliott, Jr.ca Chemical Engineering Department, Colorado School of Mines, 1500 Illinois Street, Golden, CO 80401-1887, USA

b Department of Chemical Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, UKc Chemical Engineering Department, The University of Akron, Akron, OH 44325-3906, USA

Received 27 February 2001; accepted 9 January 2002

Abstract

We develop a crossover equation of state (EOS) for square-well fluids with varying well width. This equation yieldsthe exact second and third virial coefficients, and accurately reproduces first-order (high-temperature) perturbationtheory results. In addition, this EOS yields the correct scaling exponents near the critical point. We perform extensivenew molecular dynamics and Monte Carlo simulations in the one-phase region for varying well widths ofλ = 1.5,2.0, and 3.0. We fit the parameters of our EOS to one-phase and two-phase thermodynamic data from our simulationsand those of previous researchers. The resulting EOS is found to represent the thermodynamic properties of thesesquare-well fluids to less than 1% deviation in internal energy and density and 0.1% deviation in vapor pressure.© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Computer simulations; Critical point; Crossover theory; Equation-of-state; Square-well fluids; Thermodynamicproperties; Vapor–liquid equilibria

1. Introduction

Because of its simplicity, the square-well fluid has long served as a model system for understandingthe behavior of real fluids. The interaction energyu(r) between two square-well molecules separated bya distancer is given by

u(r) =

+∞, for r < σ

−ε, for σ < r < λσ

0, for λσ < r

(1)

whereσ is the diameter of the hard-sphere repulsive interaction,ε is the strength of the attractive in-teraction, andλ characterizes the range of the attractive interaction. This model interaction possesses

∗ Corresponding author. Tel.:+1-303-273-3190; fax:+1-303-273-3730.E-mail address: [email protected] (S.B. Kiselev).

0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0378-3812(02)00022-5

122 S.B. Kiselev et al. / Fluid Phase Equilibria 200 (2002) 121–145

the basic interactions found in many real fluids: a short-range attractive interaction and a repulsiveexcluded volume interactions. As a result, it possesses all the complex behavior of real fluidsystems.

Due to its importance, the thermodynamic properties of the square-well fluid have been extensivelystudied using computer simulation methods in both the one-phase region [1–8] and the two-phase region[9–11]. There has also been a lot of work in trying to obtain an equation of state (EOS) for variable-widthsquare-well fluids.

One approach has been to construct empirical forms and to fit the parameters to available simulationdata [12–14]. Unfortunately, in these previous works, the empirical forms did not incorporate knowntheoretical results (e.g. the second and third virial coefficients, the scaling behavior near the critical point,etc.), and as a result their accuracy is limited.

Another approach that has been taken is to take the hard-sphere fluid as a reference system and touse perturbation theory around the high-temperature limit to obtain the properties of the square-wellfluid [12]. In this approach, the coefficients of the perturbation expansion have been obtained throughempirical fits to simulation data [2,12,15–17], approximate theories for the hard-sphere correlationfunctions [18,19], expansion around theλ → ∞ limit [20], and expansion around theλ → 1 limit[21,22].

A simpler description of the square-well fluid, based on the lattice theory of fluids, is given by thequasi-chemical approximation [4,5,8,23–27]. The advantage of these approaches is that they lead tosimple analytical forms for the free energy of the system. However, the predictions of the quasi-chemicalapproximation are not very accurate.

Integral equation approaches have also been employed with various approximate closure relations,including the Percus–Yevick [28], hypernetted chain [1], mean spherical approximation [29], and hybridmean-spherical approximation closures [30]. Although these methods are theoretically based, they stilldo not reproduce all the known behaviors of the square-well fluid.

One of the common difficulties with the theoretical approaches is that they do not properly describe thecritical region. Attempts have been made to construct crossover equations of state for square-well fluidswhich do yield nonclassical scaling behavior near the critical point [31–33]. However, in these approaches,the critical scaling exponents are only correct to first-order in the epsilon expansion, and do not yield thecorrect scaling of the caloric properties of the system. For example, the predicted isochoric heat capacitynear the critical point is not divergent (i.e.α = 0), in contrast to what is expected theoretically (i.e.α ≈ 0.110).

In this work, we attempt to construct a semi-empirical EOS for square-well fluids which will accuratelyrepresent the thermodynamic properties of variable-width square-well fluids at all state points. The formof the EOS is constructed such that it yields the exact second and third virial coefficients, and alsoreproduces the first-order high-temperature coefficient. In addition, near the critical point this EOS alsopossesses the correct scaling form.

The remainder of this paper is organized as follows. In Section 2, we describe our classical EOS forsquare-well fluids. In Section 3, we formulate a new crossover formulation. This new formulation, unlikeprevious ones, is well behaved far from the critical point, even in the infinite temperature limit. Wecombine our classical EOS with this new crossover formulation. In Section 4, we provide the detailsfor our Monte Carlo and molecular dynamics simulations for square-well fluids. In Section 5, we fit theparameters for our new crossover EOS to available simulation data for square-well fluids. In Section 6,we summarize the major findings of this paper.

S.B. Kiselev et al. / Fluid Phase Equilibria 200 (2002) 121–145 123

2. Classical free energy

The residual Helmholtz free energyAres of a square-well system can be divided into a contributionfrom the hard-sphere interactionsAHS and a contribution from the attractive interactionsAatt

Ares = AHS + Aatt (2)

For the free energy of a hard-sphere system, we use the Carnahan–Starling EOS [34], which has beenshown to accurately represent the thermodynamic properties of the hard-sphere fluid. Note that this EOSpossesses the exact second and third virial coefficients.

The behavior of the attractive contribution to the Helmholtz free energy is fairly complex, and manyforms have been proposed to describe it. To get a feeling for how to model this term, we examine thebehavior ofAatt in the low-density limit. In this regime, the Helmholtz free energy can be accuratelyrepresented by a virial series

Aatt

NkBT= Batt

2 ρ + 12B

att3 ρ2 + · · · (3)

whereBattn = Bn − BHS

n is the attractive contribution to thenth virial coefficient,Bn the nth virialcoefficient of the square-well fluid, andBHS

n is thenth virial coefficient of the hard-sphere fluid. For thesquare-well fluid, the second and third virial coefficients are known exactly [35]

B2 = BHS2 [1 − (λ3 − 1)∆] (4)

B3 = BHS3 [1 − f1(λ)∆− f2(λ)∆

2 − f3(λ)∆3] (5)

where∆ = exp[ε/(kBT )] − 1, and

f1(λ) =

0, for λ ≤ 115(λ

6 − 18λ4 + 32λ3 − 15), for 1 < λ < 2175 , for 2 < λ

(6)

f2(λ) =

0, for λ ≤ 125(λ

6 − 18λ4 + 16λ3 + 9λ2 − 8), for 1 < λ < 215(−32λ3 + 18λ2 + 48), for 2 < λ

(7)

f3(λ) =

0, for λ ≤ 165(λ

2 − 1)3, for 1 < λ < 215(5λ

6 − 32λ3 + 18λ2 + 26), for 2 < λ

(8)

In this limit, we see that each of the virial coefficients is a polynomial function of∆.Taking this into consideration, we propose the following empirical form forAatt

Aatt

NkBT= −A0 ln (1 + A1∆+ A2∆

2 + A3∆3) (9)


whereN is the number of particles in the system,kB the Boltzmann constant,T is the absolute temperatureof the system, and the functionsA0, A1, A2, andA3 are given by Pade approximants [36]

A0 = a(0)0 + a

(2)0 y2 + a

(3)0 y3 + a

(5)0 y5

1 + b(2)0 y2

(10)

A1 = 1

A0

[a(1)1 y

1 + b(1)1 y + b

(2)1 y2 + b

(3)1 y3

](11)

A2 = a(2)2 y2 + a

(3)2 y3

1 + b(1)2 y + b

(2)2 y2 + b

(3)2 y3

(12)

A3 = a(2)3 y2 + a

(3)3 y3

1 + b(1)3 y + b

(2)3 y2 + b

(3)3 y3

(13)

wherey = πσ 3ρ/6 is the packing fraction of the spheres, andρ is the number density of spheres.The application of the logarithmic form for the attractive free energy was motivated by the forms of theRedlich–Kwong and Peng–Robinson engineering equations, and by the observation that the predominantdensity dependence seems to soften for high density square-well fluids.

Expanding Eq. (9) around zero density and comparing the result with the virial series, we find thefollowing relations

a(1)1 = 4(λ3 − 1) (14)

b(1)1 = −5

4

f1(λ)

(λ3 − 1)(15)

a(2)2 = 1

a(0)0

[8(λ3 − 1)+ 1

a(0)0

5f2(λ)

](16)

a(2)3 = 1

a(0)0

5f3(λ) (17)

By using these relations for the coefficients, the resulting free energy will possess the exact second andthird virial coefficients.

At high-temperatures, the free energy can be expanded as

Aatt

NkBT= A′

1

(ε

kBT

)+ A′

2

(ε

kBT

)2

+ · · · (18)

where the coefficientsA′n can be determined from only the properties of the hard-sphere fluid.

For example, the coefficientA′1 can be obtained from the pair correlation function of the hard-sphere

systemgHS

A′1 = 2πρ

∫ λσ

σ

dr r2gHS(r) (19)


By expanding our free energy about the infinite temperature (i.e.ε/kBT = 0) limit, we find

A′1 = A0A1

In order to reproduce the correct first-order perturbation termA′1, we fit the parametersb(2)1 andb(3)1 to the

results of Eq. (19), withgHS(r) computed by the Verlet–Weis [37] procedure. This fit results in averagedeviations fromA′

1 of less than 0.2%, as discussed in Section 5.In the critical region, the dimensionless Helmholtz free energyA(T , v) = A(T , v)/kBT can be split

into the two parts

A(T , v) = �A(τ, ϕ)+ Areg(T , v) (20)

where the critical, or singular part of the free energy,�A is a function of the dimensionless temperaturedeviationτ = T/Tc −1 and the order parameterϕ, while the regular partAreg(T , v) is a smooth, analyticfunction of the temperatureT and the molar volumev = V/N . For the classical EOS, the critical part�A(τ, ϕ) is an analytic function of the variables and, asymptotically close to the critical point (where|τ | 1 and|ϕ| 1), can be represented by a Taylor expansion

�A(τ, ϕ) = a12τϕ2 + a04ϕ

4 (21)

that corresponds to the Landau, or mean-field (MF), theory of critical phenomena [38]. In simple fluids, adimensionless deviation of the densityρ or the molar volumev from its critical value (or their combination)[39], can be chosen as the order parameter, and the quantityh = (∂�A/∂ϕ)τ , conjugate to the orderparameterϕ, plays the role of the ordering field. From the conditionh = 0 atτ ≤ 0, we have that theequilibrium value of the order parameter on the coexistence curveϕcxs ∝ |τ |1/2. Forτ > 0, the conditionh = 0 corresponds to the conditionϕ = 0, the susceptibilityχτ = (∂ϕ/∂h)τ ∝ τ−1, and the singularpart of the isochoric heat capacity�CV = −(∂2�A/∂τ 2)V remains constant as|τ | → 0. Thus, for anyclassical EOS, the critical exponents

βL = 12, γL = 1, αL = 0. (22)

The MF behavior is valid only in the region where the large-scale fluctuations of the order parameterare negligibly small. The size of this region is characterized by the Ginzburg number (Gi) [40]

Gi = κ

(l

l0

)6

(23)

whereκ is a numerical constant which in general depends on the path of approach to the critical pointand the property of interest,I is an average distance between particles, andI0 is an effective radius of theinteraction between molecules. At zero ordering field (h = 0), which in the simple fluids corresponds tothe critical isochore (v = vc) atτ ≥ 0 or to the coexistence curve (v = vL,V ) atτ < 0, the Landau theoryis valid only in the temperature region

Gi |τ | 1. (24)

As the critical point is approached (T → Tc), at temperatures

|τ | Gi, (25)

the intensity of the long-range fluctuations of the order parameter anomalously increases and divergesat the critical point. As a consequence, at these temperatures the critical part of the thermodynamic


potential�A becomes a non-analytic, scaling function of the dimensionless temperatureτ and the orderparameterϕ. At zero ordering field, the asymptotic singular behavior of the thermodynamic propertiescan be described in terms of scaling laws [41,42]

ϕcxs = ±B0|τ |β(1 + B1|τ |∆1), χτ = Γ ±0 |τ |−γ (1 + Γ1|τ |∆1), �CV = A±

0 |τ |−α(1 + A1|τ |∆1)

(26)

where signs ‘±’ correspond to the liquid (+) and vapor (−) phases, the superscripts ‘±’ correspond tothe high (T > Tc) and low (T < Tc) temperature regions, the subscripts ‘0’ and ‘1’ correspond to theasymptotic and first Wegner correction terms, respectively, and the universal critical exponents

β = 0.325, γ = 1.24, α = 2 − γ − 2β = 0.110, ∆1 = 0.51. (27)

In order to describe the crossover from the MF behavior (with the critical exponents given by Eq. (22))to the asymptotic scaling behavior (given by Eq. (26)), the crossover theory should be applied [43,44].We describe this procedure in the following section.

3. Crossover free energy

A general procedure for transforming any classical EOS into the crossover form was proposed byKiselev [45]. This procedure has a theoretical foundation in the renormalization-group (RG) theory andhas been successfully applied to the cubic Patel–Teja (PT) EOS [45,46] and to the statistical associatingfluid theory (SAFT) EOS [47,48]. Following this method, we first rewrite the classical expression (20)for the Helmholtz free energy in dimensionless form

A(T , v) = �A(τ, ϕ)− ϕP0(T )+ Ares0 (T )+ A0(T ), (28)

where we use the definitionϕ = v/vc − 1. P0(T ) = P(T , vc)vc/kBT is the dimensionless pressure,Ar

0(T ) = Ares(T , vc)/kBT is the dimensionless residual part of the Helmholtz free energy along thecritical isochore, andA0(T ) = A0(T )/kBT is the dimensionless temperature-dependent part of the idealgas Helmholtz free energy. The critical part of the dimensionless Helmholtz free energy is

�A(τ, ϕ) = Ares(τ, ϕ)− Ares(τ,0)− ln (ϕ + 1)+ ϕP0(τ ), (29)

whereAres = Ares/kBT is the dimensionless residual part of the Helmholtz free energy.Secondly, we replaceτ andϕ in the critical part of the classical Helmholtz free energy�A in Eq. (28)

by the renormalized values [45]

τ = τY−α/2∆1, ϕ = ϕY (γ−2β)/4∆1, (30)

whereY is a crossover function to be specified below and the critical exponents are given by Eq. (27).Eq. (30) corresponds to the formal solution of the renormalization-group equations obtained near theactual critical point of the system with the real critical temperatureTc and critical volumevc. In general,these critical parameters do not coincide with the classical critical parametersT0c andv0c which are foundfrom the classical SW EOS (2) through the conditions

Pc = −(∂A

∂v

)T0c

,

(∂2A

∂v2

)T0c

= 0,

(∂3A

∂v3

)T0c

= 0. (31)


These equations for the SW EOS can only be solved numerically, and, in general,T0c, v0c, andP0c

are complicated functions of the parametersa(i)j andb(i)j . Therefore, in order to take into account the

difference between the classical and real critical parameters of the SW fluid, following Kiselev [45], wehave introduced in Eq. (30) the additional terms

τ = τY−α/2∆1 + (1 + τ)�τcY2(2−α)/3∆1, (32)

ϕ = ϕY (γ−2β)/4∆1 + (1 + ϕ)�ϕcY(2−α)/2∆1, (33)

where�τc = (Tc − T0c)/T0c = �Tc/T0c and�ϕc = (vc − v0c)/v0c = �vc/v0c are the dimensionlessshifts of the critical temperature and the critical volume, respectively. If the real critical parameters of thesystemTc andvc are known, then the critical shifts�τc and�ϕc are also known. Otherwise, followingKiselev and Ely [48] one can represent the critical shifts as functions of theGi

�τc = − δτGi

1 + Gi, �ϕc = − δϕGi

1 + Gi, (34)

where the coefficientsδτ andδϕ do not depend on the parameters of the intermolecular potential and forthe classical EOS of interest can be treated as constants. For this approach to be feasible,�τc 1 and�ϕc 1. Note that preliminary applications of perturbation theory in place of Eq. (9) resulted in valuesof �τc and�ϕc that were too large, motivating the need for more empirical parameters, as exhibited inEq. (9).

In order to complete transformation, one needs to add in Eq. (28) the kernel term

K(τ 2) = 1

2a20

(τ

τ + 1

)2

(Y−α/∆1 − 1), (35)

which provides the correct scaling behavior of the isochoric specific heat asymptotically close to the criticalpoint. We need to note, that the kernel term as given by Eq. (35) differs slightly from the correspondingkernel term employed earlier by Kiselev [45]. In order to avoid the unphysical divergence asτ → ∞, wereplaced the prefactorτ 2 in the kernel term in [45] with the functionτ 2/(τ + 1)2. In the critical region,atτ 1, the functionτ 2/(τ + 1)2 ∼= τ 2, and both terms practically coincide. As|τ | � 1, the crossoverfunctionY (q) → 1, soK→ 0.

The crossover functionY in Eqs. (30) and (35) can be written in parametric form [45]

Y (q) = [q/R(q)]∆1, (36)

where here, unlike the previous crossover model [45], for the functionR(q) we adopted the Pade-approximant of the numerical solution of the RG equations [49,50]

R(q) = 1 + q2

q + 1. (37)

In Eq. (36) the parametric variableq = r/Gi, wherer is a dimensionless measure of the distance fromthe critical point, which at the critical isochore coincides with the dimensionless temperature deviationτ

[43–45]. In the original Kiselev approach [45], the variabler was determined from a solution of the para-metric linear-model (LM) EOS. The LM EOS has a theoretical foundation in the renormalization-grouptheory [51], but it cannot be extended into the metastable region and represent analytically connectedvan der Waals loops. Therefore, in the present work, we find the parametric variabler from a solution


of the parametric sine-model developed recently by Fisher et al. [52] and represented in crossover formby Kiselev and Ely [48]. With account of the redefinition of the functionR(q) adopted in this work, thecrossover sine-model equation for the variableq can be written in the form(

q − τ

Gi

) [1 − p2

4b2

(1 − τ

qGi

)]= b2

(ϕ

m0Giβ

)2

Y (q)(1−2β)/∆1, (38)

wherem0 is a system-dependent parameters, andp2 = b2 = 1.359 are the universal sine-,p2-, andlinear-model,b2, parameters [48]. The linear-model crossover equation for the parametric variableq

employed earlier by Kiselev [49] is recaptured from Eq. (38) when the sine-model parameterp2 → 0.Finally, the crossover expression for the Helmholtz free energy can be written in the form

A(T , v)= Ares(τ , ϕ)− Ares(τ ,0)− ln (ϕ + 1)+ ϕP0(τ )

−�vP0(T )+ Ares0 (T )+ A0(T )−K(τ 2) (39)

where�v = v/v0c − 1 is the dimensionless deviation of the volume from the classical critical volumev0c, P0(T ) = P(T , v0c)v0c/kBT , andAr

0(T ) = Ares(T , v0c)/kBT are the dimensionless pressure andresidual part of the Helmholtz free energy along the classical critical isochorev = v0c, and the explicitexpressions for the renormalized residual partAres(τ , ϕ), and analytical functions,Ares(τ ,0) andP0(τ )

for the classical SW EOS are given in Appendix A. The crossover SW EOS can be obtained from thecrossover expression (39) by differentiation with respect to volume

P(T , v) = −(∂A

∂v

)T

= RT

v

[− v

vc

(∂A

∂ϕ

)T

+ v

v0cP0(T )+ v

vc

(∂K

∂ϕ

)T

]. (40)

where the kernel termK depends on the order parameterϕ only through the argument of the crossoverfunctionY (q).

Eqs. (28)–(39) completely determine the crossover Helmholtz free energy for the SW fluids. Far awayfrom the critical pointq � 1 (or|τ | � Gi atv = vc), the crossover functionY (q) → 1, the renormalizedtemperature and volume tend to their classical valuesτ → τ andϕ → �v, and Eq. (39) is transformedinto the classical Helmholtz free energy density (2). Asymptotically, close to the critical pointq 1 (or|τ | Gi at v = vc), the crossover functionY (q) ∼= q∆1, the parametersτ ∼= τq−(α/2) (or τ ∼= τ (2−α)/2

at v = vc) and ϕ ∼= ϕq(γ−2β)/4, the critical part of the free energy�A(τ , ϕ) is renormalized, and thecrossover Eq. (39) for Helmholtz free energy reproduces the asymptotic scaling laws (see Eq. (26)).

4. Simulation details

Molecular dynamics (MD) simulations were performed in the NVE ensemble for 1–3 ns of simulatedtime. Estimating the simulated time at each temperature requires assumptions about the mass and energyof the model potentials. The mass was taken as 16 amu and the values ofε/kB were taken as 146,73, and 19 K forλ = 1.5, 2.0, 3.0, respectively. Simulations for a given potential model along a givenisochore were initialized with FCC cells in accordance with the number of atoms (N ) simulated in eachsystem. Subsequent simulations along the same isochore were initialized from the final configuration ofthe preceding simulation with velocities rescaled to reduce the temperature. Simulated time was 3 ns fory < 0.19 in all cases exceptλ = 3.0. The long-range of the 3.0 potential resulted in slow simulations


because of the large number of neighbors. Simulated time was 1 ns fory > 0.19. The heat capacity ofthe system was estimated by regressing the values of internal energyUr with a third-order polynomialand computing the derivative. Note that the fluctuations in compressibility factor were relatively smallcompared to fluctuations in temperature or internal energy. This happens because higher local temperaturesinduce higher local pressures. Since the pressure is in the numerator of the compressibility factor andtemperature is in the denominator, their fluctuations tend to cancel.

In addition, Monte Carlo simulations were also performed for square-well fluids withλ = 1.5, 2.0, and3. In these simulations, the system was started in a perfect face-centered cubic lattice and equilibratedusing random displacements of each of the molecules. For each set of conditions, the simulations weredivided into separate runs, each consisting of 104 attempted moves per molecule. The reported propertieswere taken as the average value of the runs, and the uncertainty was estimated by the standard deviation.

The simulation results are available from the authors upon request.

5. Comparisons with simulation data

The crossover EOS for SW fluids contains the following system-dependent parameters: the originalclassical parametersa(i)j andb(i)j (i, j = 0–3) in Eq. (2), theGi, the critical amplitudea20 in the kernel

term, and the sine-model parameterm0. As mentioned above, the parametersa(1)1 , a(2)2 , a(2)3 , andb(1)1 are

directly related to the second and third virial coefficients of SW fluids and can be determined exactly (seeEqs. (14)–(17)). The coefficientsb(2)1 andb(3)1 determine the behavior of the residual internal energyUr

in the limit β∗ → 0 (or, equivalently, the coefficientA′1 in Eq. (18)).

Furthermore, since the third virial coefficients, and as a consequence the parametersa(2)2 , a(2)3 , and

b(1)1 in Eq. (2), have different analytical forms in the regionsλ ≤ 2.0 and forλ ≥ 2.0, the analytical

expressions for coefficientsb(2)1 andb(3)1 extracted from Eq. (2) also have different forms forλ ≤ 2.0andλ ≥ 2.0 regions. Therefore, in this work the coefficientsb

(2)1 andb(3)1 were represented by a Taylor

expansion

b(i)1 = b

(i)1,0 +

4∑k=1

b(i)1,k �λ

k, i = 2 and 3, (41)

where the parameter of expansion

�λ ={λ− 1.5, for λ ≤ 2.0

λ− 3.0, for λ ≥ 2.0(42)

and the coefficientsb(i)1,k for each region were found from a fit of Eq. (2) to the for hard-sphere MC simula-

tion data forA′1. The coefficientsb(i)1,k are listed in Table 1 and a comparison with the simulation data is given

in Fig. 1. In the entire density range shown in Fig. 1 the maximum deviation of the calculated values ofA′1

from the MC simulation results is less than 0.2%, which roughly corresponds to the accuracy of the data.Even after this elucidation, the CR EOS still contains adjustable parametersa

(i)j andb(i)j which can be

determined only from a fit of the CR EOS to the simulation data. If we also add to this number threecrossover parametersGi, a20, andm0, which in principle can dramatically change the thermodynamic


Table 1Coefficentsb(i)1,k in Eq. (41)

Coefficient λ ≤ 2 λ > 2

b(2)1,0 1.5093018500 0.4697011630

b(2)1,1 0.9673047240 −4.604133228

b(2)1,2 6.3087619550 −26.67612171

b(2)1,3 −0.709455318 −43.58375157

b(2)1,4 −20.27785515 −19.76736755

b(3)1,0 0.5077587980 −0.528853757

b(3)1,1 −0.771042541 5.1546638240

b(3)1,2 −20.76688587 40.508756730

b(3)1,3 −12.04372937 72.750520860

b(3)1,4 61.302467810 35.181735840

description of SW fluids, it becomes clear that obtaining a reliable statistical analysis of the crossoverbehavior requires further restrictions on the empirical parameters. In order to obtain a representativestatistical analysis of the crossover behavior of SW fluids, the number of the empirical parameters shouldbe essentially restricted.

With this in mind, we chose the SW fluid withλ = 3.0 as our first model system. SinceGi ∝ l−60 ∝ λ−6,

we expect that for thisλ theGi is small [53], classical MF behavior over the entire experimentally availableregion, and the crossover parametersa20, andm0 are statistically irrelevant. The critical amplitudea20

Fig. 1. Variation ofA1 with packing fraction for different square-well widths. Symbols correspond to Monte Carlo simulationresults and the lines represent the values calculated with the crossover EOS.


Fig. 2. Vapor pressureP ∗ as function of the temperatureT ∗ for the square-well fluid withλ = 3.0. The open circles correspondto the molecular dynamic results of Orkoulas and Panagiotopoulos [11], the crossed symbols mark the critical points, the solidlines correspond to the values calculated with the crossover EOS, the dotted–dashed lines represent the values calculated with theclassical quasi-chemical model (QCM) [8], the long-dashed lines correspond to the long-range approximation (LRA) [21,22],and the ants correspond to the values calculated with the second order thermodynamic perturbation theory (TPT2) [54].

determines the non-analytic singular behavior of the isochoric specific heat, which is not observed inthis system, and the parameterm0 effectively takes into account a variation of the prefactorκ in Eq. (23)for different definitions of theGi [42,44], which we suppose to be small for this system. Therefore, weinitially set�Tc = �vc = a20 = 0,m0 = 1, and all other parameters were found from a fit of the CR EOSto our MD and MC simulation data for the compressibility factorZ and the residual internal energyUr

in the one-phase region, and to the MC simulation VLE-data obtained by Orkoulas and Panagiotopoulos[11]. We found that theGi for this system is really very small,Gi = 2.9×10−3, and that for the adequatedescription of all simulation data for SW fluid withλ = 3.0 only nine adjustable parameters in Eq. (2)are needed. Comparisons of the crossover model with the simulation data are given in Figs. 2–4. Theaverage absolute deviation (ADD) between correlated and simulation data were less than 1% for bothinternal energy and density. For the vapor pressure, the ADD is of about 0.1% in the temperature region7.4 ≤ T ∗ ≤ T ∗

c . At low temperatures, the deviations of the vapor pressure calculated with the CR EOSfrom the MC data are increased up to 0.5%.The ants in Figs. 2 and 3 correspond to the values calculatedwith the second-order thermodynamic perturbation theory (TPT2) [54] and the dashed lines represent thevalues calculated with the classical quasi-chemical model (QCM) [8] (dotted–dashed lines) and long-rangeapproximation (LRA) [21,22] (long-dashed lines). For the vapor pressure these values are very close to theones calculated with the CR EOS, that confirms our assumption about dominant classical behavior in thissystem. As one can see, the LRA and TPT2 also yield a fairly good prediction for the saturated densities inSW fluid withλ = 3.0, while the QCM predicts the much lower critical temperature in this system and, as aconsequence, fails to reproduce the saturated liquid densities and the vapor densities in the critical region.


Fig. 3. Coexistence curve of the square-well fluid withλ = 3.0. The open symbols correspond to the molecular dynamic resultsof Orkoulas and Panagiotopoulos [11] for the system sizeL = 12 (squares) andL = 6 (triangles), and the other conditions aresame as in Fig. 2.

As our second step, we tried to redefine the parametera20 from a fit of the crossover model to theisochoric heat capacity,CV , obtained by numerical differentiation of the molecular dynamic results forUr. Our results forCV are shown in Fig. 5. Incorporation of a non-zero value fora20 in the CR EOS forthe SW fluid withλ = 3.0 does not improve description of the isochoric heat capacity far from the criticalregion and causes unphysical behavior at temperaturesτ ≈ Gi. In order to avoid this unphysical behavior,the parametera20, like theGi, should be very small. This observation confirmed our initial assumptionabout the predominantly classical behavior of this system. Hence, the parametera20 remains zero in ourbest representation of this system.

The classical behavior of the SW fluid withλ = 3.0 is, in our opinion, a major reason why the criticaltemperatureT ∗

c = 9.96 obtained from a solution of Eq. (31) for this system differs fromT ∗c = 9.87

obtained by Orkoulas and Panagiotopoulos [11] from the analysis of their MC simulation results. Theiranalysis was performed on the basis of the asymptotic scaling laws with the non-classical critical exponentsas given by Eq. (27). However, it is valid only in the temperature range as given by Eq. (25). Our analysisshows, that actually all VLE-data in the critical region (τ 1) for the SW fluid withλ = 3.0 presentedin [11] belong to the MF region as given by Eq. (24) withβ = 0.5, rather than to the scaling region givenby Eq. (25). As a consequence, the real critical temperature for this system is slightly higher than thecritical temperature obtained by Orkoulas and Panagiotopoulos [11] from the analysis of the VLE-datawith β = 0.325.

The second system considered here is the SW fluid withλ = 1.5. For the classical SW EOS, we usedthe same number of adjustable parametersa

(i)j andb(i)j as in the previous case forλ = 3.0. All these

parameters together with the crossover parametersGi, a20, andm0 were found from a fit of the CR EOS


Fig. 4. Compressibility factor (a) and residual internal energy (b) along various isochores as functions of the inverse temperatureβ∗ = 1/T ∗ for the square-well fluid withλ = 3.0. The symbols correspond to the Monte Carlo (empty circles) and moleculardynamic (filled circles) results of this work forN = 500, and the lines represent the values calculated with the crossover EOS.

to our(Z, y, T ), (Ur, y, T ), and(CV , y, T ) data in the one-phase region, and VLE-data presented in [11]at condition�Tc = �vc = 0. We found that the best description of all data is achieved with the crossoverparameters

Gi = 0.629, a20 = 4.059, m0 = 0.852. (43)

In contrast to the previous case forλ = 3.0 , theGi for the SW fluid withλ = 1.5 is not small. Hence,the crossover contribution is essential in the region|τ | ≤ 1. As a consequence, the critical temperaturefor this system obtained from a solution of Eq. (31) with the optimized values of the parametersa

(i)j and

b(i)j , T ∗

c = 1.244, appears to be very close toT ∗c = 1.218 obtained by Orkoulas and Panagiotopoulos [11]

from the analysis of their MD VLE-data withβ = 0.325. The critical temperature reported by Orkoulasand Panagiotopolous [11] has been obtained as extrapolation of the critical temperatures for the systemsizesL = 6, 8, and 12 to the infinite system withL → ∞, while ourT ∗

c is a critical temperature of thereal finite-size system.


Fig. 5. Isochoric heat capacity along various isochores as function of the inverse temperatureβ∗ = 1/T ∗ for the square-wellfluid with λ = 3.0. The symbols correspond to the values obtained by numerical differentiation of the molecular dynamic resultsfor Ur presented in Fig. 4, and the lines represent the values calculated with the crossover EOS witha20 = 0 (solid lines) andwith a20 = 11.5 (dashed lines).

We have also analyzed the finite-size effect of the residual internal energyUr and the isochoric heatcapacity for this system. The results of our analysis on the near-critical isochorey = 0.16 are shownin Figs. 10 and 11. The dashed curves in both figures represent the values calculated with the classicalEOS (i.e. with theGi and the critical amplitudea20 set to zero). The classical EOS gives qualitativelywrong behavior for the residual internal energy and the isochoric heat capacity in the critical region,while the CR EOS yields a very good representation of the simulation data up to the inverse temperatureβ∗ = 0.75. However, in the vicinity of the critical point at 0.75 ≤ β∗ ≤ 0.8 (or |τ | ≤ 7 × 10−2),the finite-size effect becomes essential, and all simulationCV data lie in this region below the valuescalculated with the CR EOS. Even for the system withN = 2048 particles, the largest system consideredin this work, the systematic deviations from the asymptotic scaling behavior appear at�T ≤ 0.03 (or|τ | ≤ 3 × 10−2), which is the accuracy of the determination of the critical temperature from these data.It is interesting to note, that unlike the experimental data for real fluids, the simulation results do notshow any jumps inUr or CV as the coexistence curve is crossed. They just smoothly penetrate into themetastable and even unstable regions without exhibiting any anomalies. For real fluids calorimetric datacould be used to improve the accuracy of the critical temperature (see, for example, [55,56]), but the lackof divergence for simulation data of finite systems precludes application of that approach in the presentinstance.

In the next step, we used, for the critical shifts, Eq. (34) and redefined the coefficientsa(i)j andb(i)j from a

fit of the CR EOS to the same dataset in the temperature rangeβ∗ ≤ 0.75 but with the critical temperature


Fig. 6. Vapor pressureP ∗ as function of the temperatureT ∗ for the square-well fluid withλ = 1.5. The symbols correspondto the molecular dynamic results of Orkoulas and Panagiotopoulos [11] (empty squares) and of Elliott and Hu [10] (filled circles),the crossed symbols mark the critical points for each dataset, and the short-dashed lines correspond to the values calculated withthe classical Bokis–Donohue model [14], and the other conditions are same as in Fig. 2.

Fig. 7. Coexistence curve of the square-well fluid withλ = 1.5. The open symbols correspond to the molecular dynamic resultsof Orkoulas and Panagiotopoulos [11] for the system sizeL = 12 (empty diamonds) andL = 6 (empty triangles), and the otherconditions are same as in Fig. 6.


T ∗c = 1.218 and the critical densityyc = 1.623, as proposed by Orkoulas and Panagiotopoulos [11]. With

the optimized values of the coefficientsa(i)j andb(i)j we found forλ = 1.5 the classical critical parametersT ∗

c0 = 1.222 andyc0 = 1.554, which in this case do not coincide with the real critical parametersT ∗c and

yc, and, with theGi as given in Eq. (43), the coefficients

δτ = 7.82× 10−3, δϕ = 1.11× 10−1. (44)

Comparisons of the crossover model with the simulation data for the SW fluid withλ = 1.5 are givenin Figs. 6–9. Excellent agreement of the predictions of the CR EOS with simulation data is observed,while all classical EOS fail to reproduce the vapor pressure and saturated densities for this system inthe critical region (see Figs. 6 and 7). The average deviation of the values calculated with the CR EOS

Fig. 8. Compressibility factor (a) and residual internal energy (b) along various isochores as functions of the inverse temperatureβ∗ = 1/T ∗ for the square-well fluid withλ = 1.5. The symbols correspond to the molecular dynamic results of Alder et al. [58](empty squares), of this work forN = 500 (filled triangles) andN = 2048 (filled circles), and to the Monte Carlo results of thiswork forN = 500 (empty circles), and the lines represent the values calculated with the crossover EOS.


Fig. 9. Isochoric heat capacity along various isochores as function of the inverse temperatureβ∗ = 1/T ∗ for the square-wellfluid with λ = 1.5. The empty symbols correspond to the Monte Carlo results of this work forN = 500, the filled symbolscorrespond to the values obtained by numerical differentiation of the molecular dynamic results forUr presented in Fig. 8, andthe lines represent the values calculated with the crossover EOS.

from simulation results is less than 1% for all properties shown in Figs. 7–11. For the vapor pressure, theaverage deviation is less than 0.1% (see Fig. 6).

The third system which we considered in this work is the SW fluid withλ = 2.0. Unlike the previoussystems considered in this work, the VLE properties of the SW fluid withλ = 2.0 have not been ascarefully studied, and the actual critical temperature and density were not established with the sameaccuracy as for the SW fluid withλ = 1.5. As was pointed out earlier by Kiselev and Ely [48], thecoefficientsδτ andδϕ in Eq. (34) do not depend on theGi. Therefore, in order to avoid over fitting, weused the same coefficientsδτ andδϕ in Eq. (34) for critical shifts for the SW fluid withλ = 2.0 as forλ = 1.5 (see Eq. (44)), and only theGi was treated as an adjustable parameter. As we mentioned earlier,the parameterm0 just renormalizes the numerical prefactorκ in Eq. (23) which depends on the path ofapproach to the critical point in the definition of theGi, but not on the parameters of the intermolecularpotential. Therefore, for this parameter, we also adopted the same value as obtained before forλ = 1.5(see Eq. (43)), while all other parameters, including the crossover parameters

Gi = 0.488, a20 = 3.796 (45)

were found by fitting the CR EOS to our(Z, y, T ) and (Ur, y, T ) data in the one-phase region, andVLE-data obtained by Elliott and Hu [10].

Fig. 12 shows the compressibility factor and residual internal energy along separate isochores asfunctions of the inverse temperatureβ∗ = 1/T ∗. Good agreement between calculated values and


Fig. 10. Finite-size effect of the residual internal energy along the isochorey = 0.16 for the square-well fluid withλ = 1.5. Theempty symbols correspond to the Monte Carlo results of this work forN = 108 (squares),N = 500 (triangles), andN = 2048(circles), the filled symbols correspond to the molecular dynamic results of this work for the sameN , and the lines represent thevalues calculated with the classical (dashed lines) and crossover (solid lines) EOS.

simulation data for both properties is observed. Deviations between the calculated values and simu-lation data in Fig. 12 are small, but at high densities (y > 2yc) the isochores for the residual internalenergy calculated with the CR EOS exhibit some waviness, which is physically unattractive. We do nothave a complete phase diagram for the SW fluid withλ = 2.0 and, therefore, do not know for cer-tain the location of the solid–liquid coexistence curve in this system. But if we compareλ = 2.0 fluidwith the phase diagram for the Lennard–Jones (LJ) fluids obtained by Vliegenthart and Lekkerkerker[57] (see Fig. 1 in [57]), we can also assume that, similar to the LJ fluids, in the SW fluid withλ =2.0 all data withT ∗ < T ∗

c and y > 2yc shown in Fig. 12 belong to the metastable liquid or solidphase. Future analysis of the 2.0 fluid should focus on enforcing consistency of the CR EOS withsimulation data at high density, in addition to generating more simulation data along the coexistencecurve.

The vapor pressure and coexistence curve simulation results obtained for the SW fluid withλ = 2.0by Vega et al. [9] and by Elliott and Hu [10] are shown in Figs. 13 and 14. In the critical region,there is an obvious discrepancy between the two datasets. Vega et al. [9] analyzed their simulation datafor the coexistence densities with the effective critical exponentβ = 0.53, which is very close to theMF valueβL = 0.5, while Elliott and Hu [10] used the non-classical valueβ = 0.325. As a conse-quence, the critical temperature extracted by Vega et al. [9] from their simulation data,T ∗

c = 2.764,is higher than the critical temperature obtained by Elliott and Hu [10],T ∗

c = 2.61. TheGi for thissystem,Gi = 0.488, is slightly smaller that theGi for the SW fluid withλ = 1.5 (see Eq. (43)),


Fig. 11. Finite-size effect of the isochoric heat capacity along the isochorey = 0.16 for the square-well fluid withλ = 1.5.The filled symbols correspond to the values obtained by numerical dedifferentiation of the molecular dynamic results forUr

presented in Fig. 10 and the other conditions are same as in Fig. 10.

but bigger than the valueGi = 0.111 obtained from Eq. (23) withl0/l = λ = 2.0 andGi(λ=1.5) =0.629. Thus, for the SW fluid withλ = 2.0 the Gi (Gi = 0.488), is not small enough to provide amean-field behavior in the critical region. Therefore, we would expect the critical temperature obtainedby Elliott and Hu [10] to be closer to the actual value. However, the critical temperature calculatedwith CR EOS (i.e.T ∗

c = 2.702) lies between the values obtained by Vega et al. [9] (T ∗c = 2.764)

and by Elliott and Hu [10] (T ∗c = 2.61), while the critical density calculated with CR EOS (yc =

1.394) practically coincide with the critical density obtained by Elliott and Hu [10] (yc = 1.40). Sim-ilar to observations forλ = 3.0, application ofβ = 0.50 to the coexistence data of Elliott and Huresults in closer agreement in the critical temperature with the CR EOS but with a smaller criticaldensity.

In order to estimate the influence of theGi on the values of the critical parameters for SW fluid withλ = 2.0, we fit the CR EOS to the same dataset but with fixed value ofGi = 0.112. The result of thesecalculations is shown in Figs. 13 and 14 by dashed-dotted curves. In the region where the simulationdata exist, both fits (Gi = 0.488 andGi = 0.112) give practically identical results. The major differencebetween these two fits is in the critical region, where the CR EOS withGi = 0.112 predicts a criticaltemperature,T ∗

c = 2.787, which is very close to the valueT ∗c = 2.764 obtained by Vega et al. [9]. In

Fig. 15, we compare the calculated values of the isochoric heat capacity with the values obtained bynumerical differentiation of the molecular dynamics results forUr (see Fig. 12). The solid lines in Fig. 15represent the values calculated withGi = 0.488, dashed-dotted lines withGi = 0.112, and the dashedlines correspond to classical EOS obtained from the CR EOS by settingGi = a20 = 0. For the SW fluidwith λ = 2.0, similar to the SW fluid withλ = 1.5, the CR EOS with small values of theGi (Gi = 0


Fig. 12. Compressibility factor (a) and residual internal energy (b) along various isochores as functions of the inverse temperatureβ∗ = 1/T ∗ for the square-well fluid withλ = 2.0. The symbols correspond to the molecular dynamic results of this work forN = 500 and the lines represent the values calculated with the crossover EOS.

or Gi = 0.112) are unable to even qualitatively describe isochoric heat capacity in the critical region. Inorder to reproduce quantitatively the asymptotic singular behavior of the isochoric heat capacity in thecritical region theGi for the SW fluid withλ = 2.0, Gi = 0.488, should be the same order of magnitudeas for the SW fluid withλ = 1.5. This conclusion is inconsistent with our estimates obtained with Eq. (23)for an ideal van der Waals fluid withl0/l = λ, but is in a qualitative agreement with the independentestimates of theGi for the square-well fluids obtained by Brilliantov [53].

All system-dependent coefficients in the CR EOS for SW fluids withλ = 1.5, 2.0, and 3.0 are listedin Tables 2 and 3. In Table 3, the critical parameters for SW fluids withλ = 1.5, 2.0, and 3.0 calculatedwith the CR EOS are compared with the critical parameters obtained by other investigators.

The computer program with the CR EOS for SW fluids withλ = 1.5, 2.0, and 3.0 is available from theauthors upon request.


Fig. 13. Vapor pressureP ∗ as function of the temperatureT ∗ for the square-well fluid withλ = 2.0. The symbols correspond tothe molecular dynamic results of Vega et al. [9] (empty circles) and of Elliott and Hu [10] (filled circles), the crossed symbolsmark the critical points for each dataset. The lines represent the values calculated with the crossover EOS withGi = 0.488 (solidlines) and withGi = 0.112 (dotted–dashed lines), with the long-range approximation (LRA) [21,22] (long-dashed lines), andthe ants correspond to the values calculated with the second-order thermodynamic perturbation theory (TPT2) [54].

Fig. 14. Coexistence curve of the square-well fluid withλ = 2.0. Other conditions are same as in Fig. 13.


Fig. 15. Isochoric heat capacity along various isochores as function of the inverse temperatureβ∗ = 1/T ∗ for the square-wellfluid with λ = 2.0. The symbols correspond to the values obtained by numerical differentiation of the molecular dynamic resultsfor Ur presented in Fig. 12 and the lines represent the values calculated with the crossover EOS withGi = 0 (long-dashed lines),with Gi = 0.488 (solid lines) and withGi = 0.112 (dotted–dashed lines).

Table 2System-dependent parameters for the crossover EOS

Coefficient λ = 1.5 λ = 2.0 λ = 3.0

Classical parametersa(0)0 1.86937E+00 3.00299E+00 4.32527E+00

a(2)0 −1.09222E+01 1.54331E+01 5.44662E+01

a(3)0 5.15214E+01 2.63444E+02 −1.20356E+01

a(5)0 −1.17971E+02 −2.02965E+01 −5.25380E−01

b(2)0 −5.89252E−01 2.88015E+01 7.68267E+00

a(3)2 3.44813E+01 1.81800E+02 1.32461E+02

b(2)2 −1.29119E+01 −2.17453E+01 5.38847E−01

b(3)2 5.13234E+01 1.29514E+02 8.63347E+00

a(3)3 −1.10254E+01 −9.05033E+01 −1.18665E+03

Crossover parametersGi 0.62856 0.48814 2.9 × 10−3

a20 4.05865 3.79583 0


Table 3Critical parameterts for SW fluids

λ Source T ∗c yc P ∗

c

1.5 This work 1.218 0.162 0.097[11] 1.218 0.162 0.095[10] 1.270 0.160 0.110[9] 1.219 0.157 0.108[59] 1.215 0.167 0.097

2.0 This work 2.702 0.139 0.208[10] 2.610 0.140 0.170[9] 2.764 0.118 0.197[59] 2.671 0.134 0.191[60] 2.684 0.123

3.0 This work 9.961 0.133 0.889[11] 9.871 0.135 0.841

6. Conclusions

We have performed extensive molecular dynamics and Monte Carlo simulations for square-well fluidsover a range of values forλdeveloped a new semi-empirical crossover EOS for variable-width square-wellfluids. Unlike previous equations of state, our model possess the exact second and third virial coefficients,reproduces the first high-temperature perturbation coefficient, and exhibits the correct scaling behaviorin the critical regime. The parameters of our EOS were fit to our new simulation data, as well as the dataof previous workers. The resulting EOS was found to provide an extremely accurate description of thethermodynamic properties of the systems we examined.

Acknowledgements

The research at the Colorado School of Mines was supported by the U.S. Department of Energy, Officeof Basic Energy Sciences, under Grant no. DE-FG03-95ER41568. One of the authors (L.L.) thanks theNational Institute of Standards and Technology for its financial support and hospitality.

Appendix A

The renormalized residual part of the Helmholtz free energy

Ares(τ , ϕ) = 4y − 3y2

(1 − y)2− A0(y) ln [1 + A1(y)∆+ A2(y)∆

2 + A3(y)∆3] (A.1)

is obtained from Eqs. (2)–(17) by replacement∆ → ∆ = exp{ε/[kBT0cτ + 1]} − 1 andy → y =y0c/(ϕ + 1), wherey0c = πσ 3/6v0c. The analytic function

Ares(τ ,0) = 4y0c − 3y20c

(1 − y0c)2 − A0(y0c) ln [1 + A1(y0c)∆+ A2(y0c)∆

2 + A3(y0c)∆3] (A.2)


is obtained by settingy = y0c in Eq. (A.1). The analytic function

P0(τ ) = 1 −(∂Ares

∂ϕ

)τ (ϕ=0)

= 1 + y0c

(∂Ares(τ ,0)

∂y0c

)τ

(A.3)

is given by

P0(τ )= 1 + y0c + y20c − y3

0c

(1 − y0c)3− y0c

dA0

dy0cln (1 + A1∆+ A2∆

2 + A3∆3)

−A0

(dA1

dy0c∆+ dA2

dy0c∆2 + dA3

dy0c∆3

)(1 + A1∆+ A2∆

2 + A3∆3)−1 (A.4)

where

dA0

dy0c= 2a(2)0 y0c + 3a(3)0 y2

0c + 5a(5)0 y40c

1 + b(2)0 y2

0c

− 2b(2)0

a(0)0 y0c + a

(2)0 y3

0c + a(3)0 y4

0c + a(5)0 y5

0c

(1 + b(2)0 y2

0c)2

(A.5)

dA1

dy0c= 1

A20

dA0

dy0c

[a(1)1 y0c + a

(2)1 y2

0c + a(3)1 y3

0c

1 + b(1)1 y0c + b

(2)1 y2

0c + b(3)1 y3

0c

]+ 1

A0

[a(1)1 + 2a(2)1 y0c + 3a(3)1 y2

0c

1 + b(1)1 y0c + b

(2)1 y2

0c + b(3)1 y3

0c

]

− 1

A0

[a(1)1 y0c + a

(2)1 y2

0c + a(3)1 y3

0c

(1 + b(1)1 y0c + b

(2)1 y2

0c + b(3)1 y3

0c)

](b

(1)2 + 2b(2)2 y0c + 3b(3)2 y2

0c) (A.6)

dA2

dy0c= 2a(2)2 y0c + 3a(3)2 y2

0c

1 + b(1)2 y0c + b

(2)2 y2

0c + b(3)2 y3

0c

− a(2)2 y2

0c + a(3)2 y3

0c

(1 + b(1)2 y0c + b

(2)2 y2

0c + b(3)2 y3

0c)2

× (b(1)2 + 2b(2)2 y0c + 3b(3)2 y2

0c) (A.7)

dA3

dy0c= 2a(2)3 y0c + 3a(3)3 y2

0c

1 + b(1)3 y0c + b

(2)3 y2

0c + b(3)3 y3

0c

− a(2)3 y2

0c + a(3)3 y3

0c

(1 + b(1)3 y0c + b

(2)3 y2

0c + b(3)3 y3

0c)

× (b(1)3 + 2b(2)3 y0c + 3b(3)3 y2

0c) (A.8)

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computer simulations and crossover equation of state - inside mines

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